Understanding Symbolic Logic, Virginia Klenk, 5th edition, Pearson Prentice Hall, 2008, Unit 20, Ex. 2(c), p.381

The task: construct the proof for the following argument.

1. (∃x){Ax• (y)(Ay ⊃ x=y) • (∃z)[Bz • (w)(Bw ⊃z=w) • z=x]}
2. Ba
3. ∴(x)(Ax ⊃ x=a)
4. * Ax ......... ACP
5. * Am • (y)(Ay ⊃ m=y) • (∃z)[Bz • (w)(Bw ⊃ z=w) • z=m]} ......... 1 EI x/m
6. * (y)(Ay ⊃ m=y) ......... 5 Simp.
7. * Ax ⊃ m=x ......... 6 UI y/x
8. * m=x ......... 4,7 MP
9. * (∃z)[Bz • (w)(Bw ⊃ z=w) • z=m] ......... 5 Simp.
10. * Br • (w)(Bw ⊃ r=w) • r=m ......... 9 EI z/r
11. * (w)(Bw ⊃ r=w) ......... 10 Simp.
12. * Ba ⊃ r=a ......... 11 UI w/a
13. * r = a ......... 2,12 MP
14. * r = m ......... 10 Simp.
15. * m = r ......... 14 Comm.
16. * m = a ......... 15,13 Id
17. * x = m ......... 8 Comm.
18. * x = a ......... 17,16 Id
19. Ax ⊃ x=a ......... 4-18 CP
20. (x)(Ax ⊃ x=a) ......... 19 UG

Understanding Symbolic Logic, Virginia Klenk, 5th edition, Pearson Prentice Hall, 2008, Unit 20, Ex. 2(a), p.381

Construct a proof for the following argument:

1. (x){{Fx • (∃y)[Fy • ¬ (x = y)]} ⊃(Axb ∨Abx)}
2. Fa • Fb
3. Ga • ¬ Gb
4. ∴ Aab ∨Aba
5. ¬ (a = b) ......... 3 Id
6. Fb ......... 2 Simp.
7. Fb • ¬ (a = b) ......... 6,5 Conj.
8. (∃y)[Fy • ¬ (a = y)] ......... 7 EG
9. {Fa • (∃y)[Fy • ¬ (a = y)]} ⊃(Aab ∨Aba) ......... 1 UI x/a
10. Fa ......... 2 Simp.
11. Fa • (∃y)[Fy • ¬ (a = y)] ......... 10,8 Conj.
12. Aab ∨Aba ......... 11,9 MP

The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.4E, 8(f), p. 558

The task: show that the following argument is valid,

1. (x)(Px ⊃ Qx)
2. ∴ [(∃x)Px • (∃x)Qx)] ≡ (∃x)(Px • Qx)
3. * (∃x)Px • (∃x)Qx) ......... ACP
4. * (∃x)Px ......... 3 Simp.
5. * Pa ......... 4 EI x/a
6. * Pa ⊃ Qa ......... 1 UI x/a
7. * Qa ......... 5,6 MP
8. * Pa • Qa ......... 5,7 Conj.
9. * (∃x)(Px • Qx) ......... 8 EG
10. [(∃x)Px • (∃x)Qx)] ⊃ (∃x)(Px • Qx) ......... 3-9 CP
11. * (∃x)(Px • Qx) ......... ACP
12. * Pm • Qm ......... 11 EI x/m
13. * Pm ......... 12 Simp.
14. * (∃x)Px ......... 13 EG
15. * Qm ......... 12 Simp.
16. * (∃x)Qx ......... 15 EG
17. * (∃x)Px • (∃x)Qx ......... 14,16 Conj.
18. (∃x)(Px • Qx) ⊃[(∃x)Px • (∃x)Qx)] ......... 11-17 CP
19. {[(∃x)Px • (∃x)Qx)] ⊃ (∃x)(Px • Qx)} • {(∃x)(Px • Qx) ⊃[(∃x)Px • (∃x)Qx)]} ......... 10,18 Conj.
20. [(∃x)Px • (∃x)Qx)] ≡ (∃x)(Px • Qx) ......... 19 BE

The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.4E, 8(d), p. 558

Show that the following argument is valid.

1. (x)[(Fx • Gx) ≡ (∃y)(Axy • Py)]
2. (∃x)(∃y)(Fx • Axy • Py)
3. ∴(∃x)(Fx • Gx)
4. (∃y)(Fa • Aay • Py) ......... 2 EI x/a
5. Fa • Aam • Pm ......... 4 EI y/m
6. (Fa • Ga) ≡ (∃y)(Aay • Py) ......... 1 UI x/a
7. [(Fa • Ga) ⊃ (∃y)(Aay • Py)] • [(∃y)(Aay • Py) ⊃(Fa • Ga)] ......... 6 BE
8. Aam • Pm ......... 5 Simp.
9. (∃y)(Aay • Py) ......... 8 EG
10. (∃y)(Aay • Py) ⊃(Fa • Ga) ......... 7 Simp.
11. Fa • Ga ......... 9,10 MP
12. (∃x)(Fx • Gx) ......... 11 EG